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连续型均匀分布 .
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连续型均匀分布
概率密度函数
累积分布函数
参数
a
,
b
∈
(
−
∞
,
∞
)
{\displaystyle a,b\in (-\infty ,\infty )\,\!}
值域
a
≤
x
≤
b
{\displaystyle a\leq x\leq b\,\!}
概率密度函数
1
b
−
a
for
a
≤
x
≤
b
0
f
o
r
x
<
a
o
r
x
>
b
{\displaystyle {\begin{matrix}{\frac {1}{b-a))&{\mbox{for ))a\leq x\leq b\\\\0&\mathrm {for} \ x<a\ \mathrm {or} \ x>b\end{matrix))\,\!}
累积分布函数
0
for
x
<
a
x
−
a
b
−
a
for
a
≤
x
<
b
1
for
x
≥
b
{\displaystyle {\begin{matrix}0&{\mbox{for ))x<a\\{\frac {x-a}{b-a))&~~~~~{\mbox{for ))a\leq x<b\\1&{\mbox{for ))x\geq b\end{matrix))\,\!}
期望
a
+
b
2
{\displaystyle {\frac {a+b}{2))\,\!}
中位数
a
+
b
2
{\displaystyle {\frac {a+b}{2))\,\!}
众数
任何
[
a
,
b
]
{\displaystyle [a,b]\,\!}
内的值 方差
(
b
−
a
)
2
12
{\displaystyle {\frac {(b-a)^{2)){12))\,\!}
偏度
0
{\displaystyle 0\,\!}
峰度
−
6
5
{\displaystyle -{\frac {6}{5))\,\!}
熵
ln
(
b
−
a
)
{\displaystyle \ln(b-a)\,\!}
矩生成函数
e
t
b
−
e
t
a
t
(
b
−
a
)
{\displaystyle {\frac {e^{tb}-e^{ta)){t(b-a)))\,\!}
特征函数
e
i
t
b
−
e
i
t
a
i
t
(
b
−
a
)
{\displaystyle {\frac {e^{itb}-e^{ita)){it(b-a)))\,\!}
连续型均匀分布 (英语:continuous uniform distribution )或矩形分布 (rectangular distribution )的随机变量
X
{\displaystyle {\mathit {X))}
,在其值域之内的每个等长区间上取值的概率皆相等。其概率密度函数 在该变量的值域内为常数。若
X
{\displaystyle X}
服从
[
a
,
b
]
{\displaystyle [a,b]}
上的均匀分布,则记作
X
∼
U
[
a
,
b
]
{\displaystyle X\sim U[a,b]}
。
定义
一个均匀分布在区间[a,b]上的连续型随机变量
X
{\displaystyle X}
可给出如下函数:
概率密度函数 :
f
(
x
)
=
{
1
b
−
a
for
a
≤
x
≤
b
0
elsewhere
{\displaystyle f(x)=\left\((\begin{matrix}{\frac {1}{b-a))&\ \ \ {\mbox{for ))a\leq x\leq b\\0&{\mbox{elsewhere))\end{matrix))\right.}
累积分布函数 :
F
(
x
)
=
{
0
for
x
<
a
x
−
a
b
−
a
for
a
≤
x
<
b
1
for
x
≥
b
{\displaystyle F(x)=\left\((\begin{matrix}0&{\mbox{for ))x<a\\{\frac {x-a}{b-a))&\ \ \ {\mbox{for ))a\leq x<b\\1&{\mbox{for ))x\geq b\end{matrix))\right.}
MGF:
M
X
(
t
)
=
E
(
e
t
x
)
=
e
t
b
−
e
t
a
t
(
b
−
a
)
{\displaystyle M_{X}(t)=E(e^{tx})={\frac {e^{tb}-e^{ta)){t(b-a)))}
公式
期望 和中值 :
是指连续型均匀分布函数的期望和中值等于区间[a,b]上的中间点。
E
[
X
]
=
a
+
b
2
{\displaystyle E[X]={\frac {a+b}{2))}
方差 :
V
A
R
[
X
]
=
(
b
−
a
)
2
12
{\displaystyle VAR[X]={\frac {(b-a)^{2)){12))}
均匀分布具有下属意义的等可能性。若
X
∼
U
[
a
,
b
]
{\displaystyle X\sim U[a,b]}
,则X落在[a,b]内任一子区间[c,d]上的概率 :
P
(
c
≤
x
≤
d
)
=
F
(
d
)
−
F
(
c
)
=
∫
c
d
1
b
−
a
d
x
=
d
−
c
b
−
a
{\displaystyle P(c\leq x\leq d)=F(d)-F(c)=\int _{c}^{d}{\frac {1}{b-a))\,dx={\frac {d-c}{b-a))}
只与区间[c,d]的长度有关,而与它的位置无关。
离散单变量
有限支集 无限支集
beta negative binomial
Borel
Conway–Maxwell–Poisson
discrete phase-type
Delaporte
extended negative binomial
Flory–Schulz
Gauss–Kuzmin
几何分布
对数分布
mixed Poisson
负二项分布
Panjer
parabolic fractal
泊松分布
Skellam
Yule–Simon
zeta
连续单变量
混合单变量
联合分布
Discrete:
Ewens
multinomial
Continuous:
狄利克雷分布
multivariate Laplace
多元正态分布
multivariate stable
multivariate t
normal-gamma
随机矩阵
LKJ
矩阵正态分布
matrix t
matrix gamma
威沙特分布
定向统计
循环单变量定向统计
圆均匀分布
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
球形双变量
Kent
环形双变量
bivariate von Mises
多变量
von Mises–Fisher
Bingham
退化分布 和奇异分布 其它
Circular
复合泊松分布
elliptical
exponential
natural exponential
location–scale
Maximum entropy
Mixture
Pearson
Tweedie
Wrapped
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